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2015 Online Math Open Problems
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(30)
30
2
Hide problems
2014-2015 Spring OMO #30
Let
S
S
S
be the value of
∑
n
=
1
∞
d
(
n
)
+
∑
m
=
1
ν
2
(
n
)
(
m
−
3
)
d
(
n
2
m
)
n
,
\sum_{n=1}^\infty \frac{d(n) + \sum_{m=1}^{\nu_2(n)}(m-3)d\left(\frac{n}{2^m}\right)}{n},
n
=
1
∑
∞
n
d
(
n
)
+
∑
m
=
1
ν
2
(
n
)
(
m
−
3
)
d
(
2
m
n
)
,
where
d
(
n
)
d(n)
d
(
n
)
is the number of divisors of
n
n
n
and
ν
2
(
n
)
\nu_2(n)
ν
2
(
n
)
is the exponent of
2
2
2
in the prime factorization of
n
n
n
. If
S
S
S
can be expressed as
(
ln
m
)
n
(\ln m)^n
(
ln
m
)
n
for positive integers
m
m
m
and
n
n
n
, find
1000
n
+
m
1000n + m
1000
n
+
m
.Proposed by Robin Park
2015-2016 Fall OMO #30
Ryan is learning number theory. He reads about the Möbius function
μ
:
N
→
Z
\mu : \mathbb N \to \mathbb Z
μ
:
N
→
Z
, defined by
μ
(
1
)
=
1
\mu(1)=1
μ
(
1
)
=
1
and
μ
(
n
)
=
−
∑
d
∣
n
d
≠
n
μ
(
d
)
\mu(n) = -\sum_{\substack{d\mid n \\ d \neq n}} \mu(d)
μ
(
n
)
=
−
d
∣
n
d
=
n
∑
μ
(
d
)
for
n
>
1
n>1
n
>
1
(here
N
\mathbb N
N
is the set of positive integers). However, Ryan doesn't like negative numbers, so he invents his own function: the dubious function
δ
:
N
→
N
\delta : \mathbb N \to \mathbb N
δ
:
N
→
N
, defined by the relations
δ
(
1
)
=
1
\delta(1)=1
δ
(
1
)
=
1
and
δ
(
n
)
=
∑
d
∣
n
d
≠
n
δ
(
d
)
\delta(n) = \sum_{\substack{d\mid n \\ d \neq n}} \delta(d)
δ
(
n
)
=
d
∣
n
d
=
n
∑
δ
(
d
)
for
n
>
1
n > 1
n
>
1
. Help Ryan determine the value of
1000
p
+
q
1000p+q
1000
p
+
q
, where
p
,
q
p,q
p
,
q
are relatively prime positive integers satisfying
p
q
=
∑
k
=
0
∞
δ
(
1
5
k
)
1
5
k
.
\frac{p}{q}=\sum_{k=0}^{\infty} \frac{\delta(15^k)}{15^k}.
q
p
=
k
=
0
∑
∞
1
5
k
δ
(
1
5
k
)
.
Proposed by Michael Kural
29
2
Hide problems
2014-2015 Spring OMO #29
Let
A
B
C
ABC
A
BC
be an acute scalene triangle with incenter
I
I
I
, and let
M
M
M
be the circumcenter of triangle
B
I
C
BIC
B
I
C
. Points
D
D
D
,
B
′
B'
B
′
, and
C
′
C'
C
′
lie on side
B
C
BC
BC
so that
∠
B
I
B
′
=
∠
C
I
C
′
=
∠
I
D
B
=
∠
I
D
C
=
9
0
∘
\angle BIB' = \angle CIC' = \angle IDB = \angle IDC = 90^{\circ}
∠
B
I
B
′
=
∠
C
I
C
′
=
∠
I
D
B
=
∠
I
D
C
=
9
0
∘
. Define
P
=
A
B
‾
∩
M
C
′
‾
P = \overline{AB} \cap \overline{MC'}
P
=
A
B
∩
M
C
′
,
Q
=
A
C
‾
∩
M
B
′
‾
Q = \overline{AC} \cap \overline{MB'}
Q
=
A
C
∩
M
B
′
,
S
=
M
D
‾
∩
P
Q
‾
S = \overline{MD} \cap \overline{PQ}
S
=
M
D
∩
PQ
, and
K
=
S
I
‾
∩
D
F
‾
K = \overline{SI} \cap \overline{DF}
K
=
S
I
∩
D
F
, where segment
E
F
EF
EF
is a diameter of the incircle selected so that
S
S
S
lies in the interior of segment
A
E
AE
A
E
. It is known that
K
I
=
15
x
KI=15x
K
I
=
15
x
,
S
I
=
20
x
+
15
SI=20x+15
S
I
=
20
x
+
15
,
B
C
=
20
x
5
/
2
BC=20x^{5/2}
BC
=
20
x
5/2
, and
D
I
=
20
x
3
/
2
DI=20x^{3/2}
D
I
=
20
x
3/2
, where
x
=
a
b
(
n
+
p
)
x = \tfrac ab(n+\sqrt p)
x
=
b
a
(
n
+
p
)
for some positive integers
a
a
a
,
b
b
b
,
n
n
n
,
p
p
p
, with
p
p
p
prime and
gcd
(
a
,
b
)
=
1
\gcd(a,b)=1
g
cd
(
a
,
b
)
=
1
. Compute
a
+
b
+
n
+
p
a+b+n+p
a
+
b
+
n
+
p
.Proposed by Evan Chen
2015-2016 Fall OMO #29
Given vectors
v
1
,
…
,
v
n
v_1, \dots, v_n
v
1
,
…
,
v
n
and the string
v
1
v
2
…
v
n
v_1v_2 \dots v_n
v
1
v
2
…
v
n
, we consider valid expressions formed by inserting
n
−
1
n-1
n
−
1
sets of balanced parentheses and
n
−
1
n-1
n
−
1
binary products, such that every product is surrounded by a parentheses and is one of the following forms:1. A "normal product''
a
b
ab
ab
, which takes a pair of scalars and returns a scalar, or takes a scalar and vector (in any order) and returns a vector. \\2. A "dot product''
a
⋅
b
a \cdot b
a
⋅
b
, which takes in two vectors and returns a scalar. \\3. A "cross product''
a
×
b
a \times b
a
×
b
, which takes in two vectors and returns a vector. \\An example of a valid expression when
n
=
5
n=5
n
=
5
is
(
(
(
v
1
⋅
v
2
)
v
3
)
⋅
(
v
4
×
v
5
)
)
(((v_1 \cdot v_2)v_3) \cdot (v_4 \times v_5))
(((
v
1
⋅
v
2
)
v
3
)
⋅
(
v
4
×
v
5
))
, whose final output is a scalar. An example of an invalid expression is
(
(
(
v
1
×
(
v
2
×
v
3
)
)
×
(
v
4
⋅
v
5
)
)
(((v_1 \times (v_2 \times v_3)) \times (v_4 \cdot v_5))
(((
v
1
×
(
v
2
×
v
3
))
×
(
v
4
⋅
v
5
))
; even though every product is surrounded by parentheses, in the last step one tries to take the cross product of a vector and a scalar. \\Denote by
T
n
T_n
T
n
the number of valid expressions (with
T
1
=
1
T_1 = 1
T
1
=
1
), and let
R
n
R_n
R
n
denote the remainder when
T
n
T_n
T
n
is divided by
4
4
4
. Compute
R
1
+
R
2
+
R
3
+
…
+
R
1
,
000
,
000
R_1 + R_2 + R_3 + \ldots + R_{1,000,000}
R
1
+
R
2
+
R
3
+
…
+
R
1
,
000
,
000
. Proposed by Ashwin Sah
28
2
Hide problems
2014-2015 Spring OMO #28
Find the number of ordered pairs
(
P
(
x
)
,
Q
(
x
)
)
(P(x),Q(x))
(
P
(
x
)
,
Q
(
x
))
of polynomials with integer coefficients such that
P
(
x
)
2
+
Q
(
x
)
2
=
(
x
4096
−
1
)
2
.
P(x)^2+Q(x)^2=\left(x^{4096}-1\right)^2.
P
(
x
)
2
+
Q
(
x
)
2
=
(
x
4096
−
1
)
2
.
Proposed by Michael Kural
2015-2016 Fall OMO #28
Let
N
N
N
be the number of
2015
2015
2015
-tuples of (not necessarily distinct) subsets
(
S
1
,
S
2
,
…
,
S
2015
)
(S_1, S_2, \dots, S_{2015})
(
S
1
,
S
2
,
…
,
S
2015
)
of
{
1
,
2
,
…
,
2015
}
\{1, 2, \dots, 2015 \}
{
1
,
2
,
…
,
2015
}
such that the number of permutations
σ
\sigma
σ
of
{
1
,
2
,
…
,
2015
}
\{1, 2, \dots, 2015 \}
{
1
,
2
,
…
,
2015
}
satisfying
σ
(
i
)
∈
S
i
\sigma(i) \in S_i
σ
(
i
)
∈
S
i
for all
1
≤
i
≤
2015
1 \le i \le 2015
1
≤
i
≤
2015
is odd. Let
k
2
,
k
3
k_2, k_3
k
2
,
k
3
be the largest integers such that
2
k
2
∣
N
2^{k_2} | N
2
k
2
∣
N
and
3
k
3
∣
N
3^{k_3} | N
3
k
3
∣
N
respectively. Find
k
2
+
k
3
.
k_2 + k_3.
k
2
+
k
3
.
Proposed by Yang Liu
27
2
Hide problems
2014-2015 Spring OMO #27
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral satisfying
∠
B
C
D
=
∠
C
D
A
\angle BCD=\angle CDA
∠
BC
D
=
∠
C
D
A
. Suppose rays
A
D
AD
A
D
and
B
C
BC
BC
meet at
E
E
E
, and let
Γ
\Gamma
Γ
be the circumcircle of
A
B
E
ABE
A
BE
. Let
Γ
1
\Gamma_1
Γ
1
be a circle tangent to ray
C
D
CD
C
D
past
D
D
D
at
W
W
W
, segment
A
D
AD
A
D
at
X
X
X
, and internally tangent to
Γ
\Gamma
Γ
. Similarly, let
Γ
2
\Gamma_2
Γ
2
be a circle tangent to ray
D
C
DC
D
C
past
C
C
C
at
Y
Y
Y
, segment
B
C
BC
BC
at
Z
Z
Z
, and internally tangent to
Γ
\Gamma
Γ
. Let
P
P
P
be the intersection of
W
X
WX
W
X
and
Y
Z
YZ
Y
Z
, and suppose
P
P
P
lies on
Γ
\Gamma
Γ
. If
F
F
F
is the
E
E
E
-excenter of triangle
A
B
E
ABE
A
BE
, and
A
B
=
544
AB=544
A
B
=
544
,
A
E
=
2197
AE=2197
A
E
=
2197
,
B
E
=
2299
BE=2299
BE
=
2299
, then find
m
+
n
m+n
m
+
n
, where
F
P
=
m
n
FP=\tfrac{m}{n}
FP
=
n
m
with
m
,
n
m,n
m
,
n
relatively prime positive integers.Proposed by Michael Kural
2015-2016 Fall OMO #27
For integers
0
≤
m
,
n
≤
64
0 \le m,n \le 64
0
≤
m
,
n
≤
64
, let
α
(
m
,
n
)
\alpha(m,n)
α
(
m
,
n
)
be the number of nonnegative integers
k
k
k
for which
⌊
m
/
2
k
⌋
\left\lfloor m/2^k \right\rfloor
⌊
m
/
2
k
⌋
and
⌊
n
/
2
k
⌋
\left\lfloor n/2^k \right\rfloor
⌊
n
/
2
k
⌋
are both odd integers. Consider a
65
×
65
65 \times 65
65
×
65
matrix
M
M
M
whose
(
i
,
j
)
(i,j)
(
i
,
j
)
th entry (for
1
≤
i
,
j
≤
65
1 \le i, j \le 65
1
≤
i
,
j
≤
65
) is
(
−
1
)
α
(
i
−
1
,
j
−
1
)
.
(-1)^{\alpha(i-1, j-1)}.
(
−
1
)
α
(
i
−
1
,
j
−
1
)
.
Compute the remainder when
det
M
\det M
det
M
is divided by
1000
1000
1000
. Proposed by Evan Chen
26
2
Hide problems
2014-2015 Spring OMO #26
Consider a sequence
T
0
,
T
1
,
…
T_0, T_1, \dots
T
0
,
T
1
,
…
of polynomials defined recursively by
T
0
(
x
)
=
2
T_0(x) = 2
T
0
(
x
)
=
2
,
T
1
(
x
)
=
x
T_1(x)=x
T
1
(
x
)
=
x
, and
T
n
+
2
(
x
)
=
x
T
n
+
1
(
x
)
−
T
n
(
x
)
T_{n+2}(x) = xT_{n+1}(x) - T_n(x)
T
n
+
2
(
x
)
=
x
T
n
+
1
(
x
)
−
T
n
(
x
)
for each nonnegative integer
n
n
n
. Let
L
n
L_n
L
n
be the sequence of Lucas Numbers, defined by
L
0
=
2
L_0 = 2
L
0
=
2
,
L
1
=
1
L_1 = 1
L
1
=
1
, and
L
n
+
2
=
L
n
+
L
n
+
1
L_{n+2} = L_n+L_{n+1}
L
n
+
2
=
L
n
+
L
n
+
1
for every nonnegative integer
n
n
n
. Find the remainder when
T
0
(
L
0
)
+
T
1
(
L
2
)
+
T
2
(
L
4
)
+
⋯
+
T
359
(
L
718
)
T_0\left( L_0 \right) + T_1 \left( L_2 \right) + T_2 \left( L_4 \right) + \dots + T_{359} \left( L_{718} \right)
T
0
(
L
0
)
+
T
1
(
L
2
)
+
T
2
(
L
4
)
+
⋯
+
T
359
(
L
718
)
is divided by
359
359
359
.Proposed by Yang Liu
2015-2016 Fall OMO #26
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
=
72
,
A
C
=
98
,
B
C
=
110
AB=72,AC=98,BC=110
A
B
=
72
,
A
C
=
98
,
BC
=
110
, and circumcircle
Γ
\Gamma
Γ
, and let
M
M
M
be the midpoint of arc
B
C
BC
BC
not containing
A
A
A
on
Γ
\Gamma
Γ
. Let
A
′
A'
A
′
be the reflection of
A
A
A
over
B
C
BC
BC
, and suppose
M
B
MB
MB
meets
A
C
AC
A
C
at
D
D
D
, while
M
C
MC
MC
meets
A
B
AB
A
B
at
E
E
E
. If
M
A
′
MA'
M
A
′
meets
D
E
DE
D
E
at
F
F
F
, find the distance from
F
F
F
to the center of
Γ
\Gamma
Γ
.Proposed by Michael Kural
25
2
Hide problems
2014-2015 Spring OMO #25
Let
V
0
=
∅
V_0 = \varnothing
V
0
=
∅
be the empty set and recursively define
V
n
+
1
V_{n+1}
V
n
+
1
to be the set of all
2
∣
V
n
∣
2^{|V_n|}
2
∣
V
n
∣
subsets of
V
n
V_n
V
n
for each
n
=
0
,
1
,
…
n=0,1,\dots
n
=
0
,
1
,
…
. For example V_2 = \left\{ \varnothing, \left\{ \varnothing \right\} \right\} \text{and} V_3 = \left\{ \varnothing, \left\{ \varnothing \right\}, \left\{ \left\{ \varnothing \right\} \right\}, \left\{ \varnothing, \left\{ \varnothing \right\} \right\} \right\}. A set
x
∈
V
5
x \in V_5
x
∈
V
5
is called transitive if each element of
x
x
x
is a subset of
x
x
x
. How many such transitive sets are there?Proposed by Evan Chen
2015-2016 Fall OMO #25
Define
∥
A
−
B
∥
=
(
x
A
−
x
B
)
2
+
(
y
A
−
y
B
)
2
\left\lVert A-B \right\rVert = (x_A-x_B)^2+(y_A-y_B)^2
∥
A
−
B
∥
=
(
x
A
−
x
B
)
2
+
(
y
A
−
y
B
)
2
for every two points
A
=
(
x
A
,
y
A
)
A = (x_A, y_A)
A
=
(
x
A
,
y
A
)
and
B
=
(
x
B
,
y
B
)
B = (x_B, y_B)
B
=
(
x
B
,
y
B
)
in the plane. Let
S
S
S
be the set of points
(
x
,
y
)
(x,y)
(
x
,
y
)
in the plane for which
x
,
y
∈
{
0
,
1
,
…
,
100
}
x,y \in \left\{ 0,1,\dots,100 \right\}
x
,
y
∈
{
0
,
1
,
…
,
100
}
. Find the number of functions
f
:
S
→
S
f : S \to S
f
:
S
→
S
such that
∥
A
−
B
∥
≡
∥
f
(
A
)
−
f
(
B
)
∥
(
m
o
d
101
)
\left\lVert A-B \right\rVert \equiv \left\lVert f(A)-f(B) \right\rVert \pmod{101}
∥
A
−
B
∥
≡
∥
f
(
A
)
−
f
(
B
)
∥
(
mod
101
)
for any
A
,
B
∈
S
A, B \in S
A
,
B
∈
S
. Proposed by Victor Wang
24
2
Hide problems
2014-2015 Spring OMO #24
Suppose we have
10
10
10
balls and
10
10
10
colors. For each ball, we (independently) color it one of the
10
10
10
colors, then group the balls together by color at the end. If
S
S
S
is the expected value of the square of the number of distinct colors used on the balls, find the sum of the digits of
S
S
S
written as a decimal.Proposed by Michael Kural
2015-2016 Fall OMO #24
Let
A
B
C
ABC
A
BC
be an acute triangle with incenter
I
I
I
; ray
A
I
AI
A
I
meets the circumcircle
Ω
\Omega
Ω
of
A
B
C
ABC
A
BC
at
M
≠
A
M \neq A
M
=
A
. Suppose
T
T
T
lies on line
B
C
BC
BC
such that
∠
M
I
T
=
9
0
∘
\angle MIT=90^{\circ}
∠
M
I
T
=
9
0
∘
. Let
K
K
K
be the foot of the altitude from
I
I
I
to
T
M
‾
\overline{TM}
TM
. Given that
sin
B
=
55
73
\sin B = \frac{55}{73}
sin
B
=
73
55
and
sin
C
=
77
85
\sin C = \frac{77}{85}
sin
C
=
85
77
, and
B
K
C
K
=
m
n
\frac{BK}{CK} = \frac mn
C
K
B
K
=
n
m
in lowest terms, compute
m
+
n
m+n
m
+
n
.Proposed by Evan Chen
23
2
Hide problems
2014-2015 Spring OMO #23
Let
N
=
12
!
N = 12!
N
=
12
!
and denote by
X
X
X
the set of positive divisors of
N
N
N
other than
1
1
1
. A pseudo-ultrafilter
U
U
U
is a nonempty subset of
X
X
X
such that for any
a
,
b
∈
X
a,b \in X
a
,
b
∈
X
:
How many such pseudo-ultrafilters are there?Proposed by Evan Chen
2015-2016 Fall OMO #23
Let
p
=
2017
,
p = 2017,
p
=
2017
,
a prime number. Let
N
N
N
be the number of ordered triples
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
of integers such that
1
≤
a
,
b
≤
p
(
p
−
1
)
1 \le a,b \le p(p-1)
1
≤
a
,
b
≤
p
(
p
−
1
)
and
a
b
−
b
a
=
p
⋅
c
a^b-b^a=p \cdot c
a
b
−
b
a
=
p
⋅
c
. Find the remainder when
N
N
N
is divided by
1000000.
1000000.
1000000.
Proposed by Evan Chen and Ashwin Sah Remark: The problem was initially proposed for
p
=
3
,
p = 3,
p
=
3
,
and
1
≤
a
,
b
≤
30.
1 \le a, b \le 30.
1
≤
a
,
b
≤
30.
22
2
Hide problems
2014-2015 Spring OMO #22
For a positive integer
n
n
n
let
n
#
n\#
n
#
denote the product of all primes less than or equal to
n
n
n
(or
1
1
1
if there are no such primes), and let
F
(
n
)
F(n)
F
(
n
)
denote the largest integer
k
k
k
for which
k
#
k\#
k
#
divides
n
n
n
. Find the remainder when
F
(
1
)
+
F
(
2
)
+
F
(
3
)
+
⋯
+
F
(
2015
#
−
1
)
+
F
(
2015
#
)
F(1) + F(2) +F(3) + \dots + F(2015\#-1) + F(2015\#)
F
(
1
)
+
F
(
2
)
+
F
(
3
)
+
⋯
+
F
(
2015#
−
1
)
+
F
(
2015#
)
is divided by
3999991
3999991
3999991
.Proposed by Evan Chen
2015-2016 Fall OMO #22
Let
W
=
…
x
−
1
x
0
x
1
x
2
…
W = \ldots x_{-1}x_0x_1x_2 \ldots
W
=
…
x
−
1
x
0
x
1
x
2
…
be an infinite periodic word consisting of only the letters
a
a
a
and
b
b
b
. The minimal period of
W
W
W
is
2
2016
2^{2016}
2
2016
. Say that a word
U
U
U
appears in
W
W
W
if there are indices
k
≤
ℓ
k \le \ell
k
≤
ℓ
such that
U
=
x
k
x
k
+
1
…
x
ℓ
U = x_kx_{k+1} \ldots x_{\ell}
U
=
x
k
x
k
+
1
…
x
ℓ
. A word
U
U
U
is called special if
U
a
,
U
b
,
a
U
,
b
U
Ua, Ub, aU, bU
U
a
,
U
b
,
a
U
,
b
U
all appear in
W
W
W
. (The empty word is considered special) You are given that there are no special words of length greater than 2015. Let
N
N
N
be the minimum possible number of special words. Find the remainder when
N
N
N
is divided by
1000
1000
1000
. Proposed by Yang Liu
21
2
Hide problems
2014-2015 Spring OMO #21
Let
A
1
A
2
A
3
A
4
A
5
A_1A_2A_3A_4A_5
A
1
A
2
A
3
A
4
A
5
be a regular pentagon inscribed in a circle with area
5
+
5
10
π
\tfrac{5+\sqrt{5}}{10}\pi
10
5
+
5
π
. For each
i
=
1
,
2
,
…
,
5
i=1,2,\dots,5
i
=
1
,
2
,
…
,
5
, points
B
i
B_i
B
i
and
C
i
C_i
C
i
lie on ray
A
i
A
i
+
1
→
\overrightarrow{A_iA_{i+1}}
A
i
A
i
+
1
such that B_iA_i \cdot B_iA_{i+1} = B_iA_{i+2} \text{and} C_iA_i \cdot C_iA_{i+1} = C_iA_{i+2}^2where indices are taken modulo 5. The value of
[
B
1
B
2
B
3
B
4
B
5
]
[
C
1
C
2
C
3
C
4
C
5
]
\tfrac{[B_1B_2B_3B_4B_5]}{[C_1C_2C_3C_4C_5]}
[
C
1
C
2
C
3
C
4
C
5
]
[
B
1
B
2
B
3
B
4
B
5
]
(where
[
P
]
[\mathcal P]
[
P
]
denotes the area of polygon
P
\mathcal P
P
) can be expressed as
a
+
b
5
c
\tfrac{a+b\sqrt{5}}{c}
c
a
+
b
5
, where
a
a
a
,
b
b
b
, and
c
c
c
are integers, and
c
>
0
c > 0
c
>
0
is as small as possible. Find
100
a
+
10
b
+
c
100a+10b+c
100
a
+
10
b
+
c
.Proposed by Robin Park
2015-2016 Fall OMO #21
Toner Drum and Celery Hilton are both running for president. A total of
2015
2015
2015
people cast their vote, giving
60
%
60\%
60%
to Toner Drum. Let
N
N
N
be the number of "representative'' sets of the
2015
2015
2015
voters that could have been polled to correctly predict the winner of the election (i.e. more people in the set voted for Drum than Hilton). Compute the remainder when
N
N
N
is divided by
2017
2017
2017
. Proposed by Ashwin Sah
20
2
Hide problems
2014-2015 Spring OMO #20
Consider polynomials
P
P
P
of degree
2015
2015
2015
, all of whose coefficients are in the set
{
0
,
1
,
…
,
2010
}
\{0,1,\dots,2010\}
{
0
,
1
,
…
,
2010
}
. Call such a polynomial good if for every integer
m
m
m
, one of the numbers
P
(
m
)
−
20
P(m)-20
P
(
m
)
−
20
,
P
(
m
)
−
15
P(m)-15
P
(
m
)
−
15
,
P
(
m
)
−
1234
P(m)-1234
P
(
m
)
−
1234
is divisible by
2011
2011
2011
, and there exist integers
m
20
,
m
15
,
m
1234
m_{20}, m_{15}, m_{1234}
m
20
,
m
15
,
m
1234
such that
P
(
m
20
)
−
20
,
P
(
m
15
)
−
15
,
P
(
m
1234
)
−
1234
P(m_{20})-20, P(m_{15})-15, P(m_{1234})-1234
P
(
m
20
)
−
20
,
P
(
m
15
)
−
15
,
P
(
m
1234
)
−
1234
are all multiples of
2011
2011
2011
. Let
N
N
N
be the number of good polynomials. Find the remainder when
N
N
N
is divided by
1000
1000
1000
.Proposed by Yang Liu
2015-2016 Fall OMO #20
Amandine and Brennon play a turn-based game, with Amadine starting. On their turn, a player must select a positive integer which cannot be represented as a sum of multiples of any of the previously selected numbers. For example, if
3
,
5
3, 5
3
,
5
have been selected so far, only
1
,
2
,
4
,
7
1, 2, 4, 7
1
,
2
,
4
,
7
are available to be picked; if only
3
3
3
has been selected so far, all numbers not divisible by three are eligible. A player loses immediately if they select the integer
1
1
1
.Call a number
n
n
n
feminist if
gcd
(
n
,
6
)
=
1
\gcd(n, 6) = 1
g
cd
(
n
,
6
)
=
1
and if Amandine wins if she starts with
n
n
n
. Compute the sum of the feminist numbers less than
40
40
40
.Proposed by Ashwin Sah
19
2
Hide problems
2014-2015 Spring OMO #19
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
=
80
,
B
C
=
100
,
A
C
=
60
AB = 80, BC = 100, AC = 60
A
B
=
80
,
BC
=
100
,
A
C
=
60
. Let
D
,
E
,
F
D, E, F
D
,
E
,
F
lie on
B
C
,
A
C
,
A
B
BC, AC, AB
BC
,
A
C
,
A
B
such that
C
D
=
10
,
A
E
=
45
,
B
F
=
60
CD = 10, AE = 45, BF = 60
C
D
=
10
,
A
E
=
45
,
BF
=
60
. Let
P
P
P
be a point in the plane of triangle
A
B
C
ABC
A
BC
. The minimum possible value of
A
P
+
B
P
+
C
P
+
D
P
+
E
P
+
F
P
AP+BP+CP+DP+EP+FP
A
P
+
BP
+
CP
+
D
P
+
EP
+
FP
can be expressed in the form
x
+
y
+
z
\sqrt{x}+\sqrt{y}+\sqrt{z}
x
+
y
+
z
for integers
x
,
y
,
z
x, y, z
x
,
y
,
z
. Find
x
+
y
+
z
x+y+z
x
+
y
+
z
.Proposed by Yang Liu
2015-2016 Fall OMO #19
For any set
S
S
S
, let
P
(
S
)
P(S)
P
(
S
)
be its power set, the set of all of its subsets. Over all sets
A
A
A
of
2015
2015
2015
arbitrary finite sets, let
N
N
N
be the maximum possible number of ordered pairs
(
S
,
T
)
(S,T)
(
S
,
T
)
such that
S
∈
P
(
A
)
,
T
∈
P
(
P
(
A
)
)
S \in P(A), T \in P(P(A))
S
∈
P
(
A
)
,
T
∈
P
(
P
(
A
))
,
S
∈
T
S \in T
S
∈
T
, and
S
⊆
T
S \subseteq T
S
⊆
T
. (Note that by convention, a set may never contain itself.) Find the remainder when
N
N
N
is divided by
1000.
1000.
1000.
Proposed by Ashwin Sah
18
2
Hide problems
2014-2015 Spring OMO #18
Alex starts with a rooted tree with one vertex (the root). For a vertex
v
v
v
, let the size of the subtree of
v
v
v
be
S
(
v
)
S(v)
S
(
v
)
. Alex plays a game that lasts nine turns. At each turn, he randomly selects a vertex in the tree, and adds a child vertex to that vertex. After nine turns, he has ten total vertices. Alex selects one of these vertices at random (call the vertex
v
1
v_1
v
1
). The expected value of
S
(
v
1
)
S(v_1)
S
(
v
1
)
is of the form
m
n
\tfrac{m}{n}
n
m
for relatively prime positive integers
m
,
n
m, n
m
,
n
. Find
m
+
n
m+n
m
+
n
.Note: In a rooted tree, the subtree of
v
v
v
consists of its indirect or direct descendants (including
v
v
v
itself).Proposed by Yang Liu
2015-2016 Fall OMO #18
Given an integer
n
n
n
, an integer
1
≤
a
≤
n
1 \le a \le n
1
≤
a
≤
n
is called
n
n
n
-well if
⌊
n
⌊
n
/
a
⌋
⌋
=
a
.
\left\lfloor\frac{n}{\left\lfloor n/a \right\rfloor}\right\rfloor = a.
⌊
⌊
n
/
a
⌋
n
⌋
=
a
.
Let
f
(
n
)
f(n)
f
(
n
)
be the number of
n
n
n
-well numbers, for each integer
n
≥
1
n \ge 1
n
≥
1
. Compute
f
(
1
)
+
f
(
2
)
+
…
+
f
(
9999
)
f(1) + f(2) + \ldots + f(9999)
f
(
1
)
+
f
(
2
)
+
…
+
f
(
9999
)
.Proposed by Ashwin Sah
17
2
Hide problems
2014-2015 Spring OMO #17
Let
A
,
B
,
M
,
C
,
D
A,B,M,C,D
A
,
B
,
M
,
C
,
D
be distinct points on a line such that
A
B
=
B
M
=
M
C
=
C
D
=
6.
AB=BM=MC=CD=6.
A
B
=
BM
=
MC
=
C
D
=
6.
Circles
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
with centers
O
1
O_1
O
1
and
O
2
O_2
O
2
and radius
4
4
4
and
9
9
9
are tangent to line
A
D
AD
A
D
at
A
A
A
and
D
D
D
respectively such that
O
1
,
O
2
O_1,O_2
O
1
,
O
2
lie on the same side of line
A
D
.
AD.
A
D
.
Let
P
P
P
be the point such that
P
B
⊥
O
1
M
PB\perp O_1M
PB
⊥
O
1
M
and
P
C
⊥
O
2
M
.
PC\perp O_2M.
PC
⊥
O
2
M
.
Determine the value of
P
O
2
2
−
P
O
1
2
.
PO_2^2-PO_1^2.
P
O
2
2
−
P
O
1
2
.
Proposed by Ray Li
2015-2016 Fall OMO #17
Let
x
1
…
,
x
42
x_1 \dots, x_{42}
x
1
…
,
x
42
, be real numbers such that
5
x
i
+
1
−
x
i
−
3
x
i
x
i
+
1
=
1
5x_{i+1}-x_i-3x_ix_{i+1}=1
5
x
i
+
1
−
x
i
−
3
x
i
x
i
+
1
=
1
for each
1
≤
i
≤
42
1 \le i \le 42
1
≤
i
≤
42
, with
x
1
=
x
43
x_1=x_{43}
x
1
=
x
43
. Find all the product of all possible values for
x
1
+
x
2
+
⋯
+
x
42
x_1 + x_2 + \dots + x_{42}
x
1
+
x
2
+
⋯
+
x
42
. Proposed by Michael Ma
16
2
Hide problems
2014-2015 Spring OMO #16
Joe is given a permutation
p
=
(
a
1
,
a
2
,
a
3
,
a
4
,
a
5
)
p = (a_1, a_2, a_3, a_4, a_{5})
p
=
(
a
1
,
a
2
,
a
3
,
a
4
,
a
5
)
of
(
1
,
2
,
3
,
4
,
5
)
(1, 2, 3, 4, 5)
(
1
,
2
,
3
,
4
,
5
)
. A swap is an ordered pair
(
i
,
j
)
(i, j)
(
i
,
j
)
with
1
≤
i
<
j
≤
5
1 \le i < j \le 5
1
≤
i
<
j
≤
5
, and this allows Joe to swap the positions
i
i
i
and
j
j
j
in the permutation. For example, if Joe starts with the permutation
(
1
,
2
,
3
,
4
,
5
)
(1, 2, 3, 4, 5)
(
1
,
2
,
3
,
4
,
5
)
, and uses the swaps
(
1
,
2
)
(1, 2)
(
1
,
2
)
and
(
1
,
3
)
(1, 3)
(
1
,
3
)
, the permutation becomes
(
1
,
2
,
3
,
4
,
5
)
→
(
2
,
1
,
3
,
4
,
5
)
→
(
3
,
1
,
2
,
4
,
5
)
.
(1, 2, 3, 4, 5) \rightarrow (2, 1, 3, 4, 5) \rightarrow (3, 1, 2, 4, 5).
(
1
,
2
,
3
,
4
,
5
)
→
(
2
,
1
,
3
,
4
,
5
)
→
(
3
,
1
,
2
,
4
,
5
)
.
Out of all
(
5
2
)
=
10
\tbinom{5}{2} = 10
(
2
5
)
=
10
swaps, Joe chooses
4
4
4
of them to be in a set of swaps
S
S
S
. Joe notices that from
p
p
p
he could reach any permutation of
(
1
,
2
,
3
,
4
,
5
)
(1, 2, 3, 4, 5)
(
1
,
2
,
3
,
4
,
5
)
using only the swaps in
S
S
S
. How many different sets are possible?Proposed by Yang Liu
2015-2016 Fall OMO #16
Given a (nondegenrate) triangle
A
B
C
ABC
A
BC
with positive integer angles (in degrees), construct squares
B
C
D
1
D
2
,
A
C
E
1
E
2
BCD_1D_2, ACE_1E_2
BC
D
1
D
2
,
A
C
E
1
E
2
outside the triangle. Given that
D
1
,
D
2
,
E
1
,
E
2
D_1, D_2, E_1, E_2
D
1
,
D
2
,
E
1
,
E
2
all lie on a circle, how many ordered triples
(
∠
A
,
∠
B
,
∠
C
)
(\angle A, \angle B, \angle C)
(
∠
A
,
∠
B
,
∠
C
)
are possible?Proposed by Yang Liu
15
2
Hide problems
2014-2015 Spring OMO #15
Let
a
a
a
,
b
b
b
,
c
c
c
, and
d
d
d
be positive real numbers such that a^2 + b^2 - c^2 - d^2 = 0 \text{and} a^2 - b^2 - c^2 + d^2 = \frac{56}{53}(bc + ad). Let
M
M
M
be the maximum possible value of
a
b
+
c
d
b
c
+
a
d
\tfrac{ab+cd}{bc+ad}
b
c
+
a
d
ab
+
c
d
. If
M
M
M
can be expressed as
m
n
\tfrac{m}{n}
n
m
, where
m
m
m
and
n
n
n
are relatively prime positive integers, find
100
m
+
n
100m + n
100
m
+
n
.Proposed by Robin Park
2015-2016 Fall OMO #15
A regular
2015
2015
2015
-simplex
P
\mathcal P
P
has
2016
2016
2016
vertices in
2015
2015
2015
-dimensional space such that the distances between every pair of vertices are equal. Let
S
S
S
be the set of points contained inside
P
\mathcal P
P
that are closer to its center than any of its vertices. The ratio of the volume of
S
S
S
to the volume of
P
\mathcal P
P
is
m
n
\frac mn
n
m
, where
m
m
m
and
n
n
n
are relatively prime positive integers. Find the remainder when
m
+
n
m+n
m
+
n
is divided by
1000
1000
1000
. Proposed by James Lin
14
2
Hide problems
2014-2015 Spring OMO #14
Let
A
B
C
D
ABCD
A
BC
D
be a square with side length
2015
2015
2015
. A disk with unit radius is packed neatly inside corner
A
A
A
(i.e. tangent to both
A
B
‾
\overline{AB}
A
B
and
A
D
‾
\overline{AD}
A
D
). Alice kicks the disk, which bounces off
C
D
‾
\overline{CD}
C
D
,
B
C
‾
\overline{BC}
BC
,
A
B
‾
\overline{AB}
A
B
,
D
A
‾
\overline{DA}
D
A
,
D
C
‾
\overline{DC}
D
C
in that order, before landing neatly into corner
B
B
B
. What is the total distance the center of the disk travelled?Proposed by Evan Chen
2015-2016 Fall OMO #14
Let
a
1
a_1
a
1
,
a
2
,
…
,
a
2015
a_2, \dots, a_{2015}
a
2
,
…
,
a
2015
be a sequence of positive integers in
[
1
,
100
]
[1,100]
[
1
,
100
]
. Call a nonempty contiguous subsequence of this sequence good if the product of the integers in it leaves a remainder of
1
1
1
when divided by
101
101
101
. In other words, it is a pair of integers
(
x
,
y
)
(x, y)
(
x
,
y
)
such that
1
≤
x
≤
y
≤
2015
1 \le x \le y \le 2015
1
≤
x
≤
y
≤
2015
and
a
x
a
x
+
1
…
a
y
−
1
a
y
≡
1
(
m
o
d
101
)
.
a_xa_{x+1}\dots a_{y-1}a_y \equiv 1 \pmod{101}.
a
x
a
x
+
1
…
a
y
−
1
a
y
≡
1
(
mod
101
)
.
Find the minimum possible number of good subsequences across all possible
(
a
i
)
(a_i)
(
a
i
)
.Proposed by Yang Liu
13
2
Hide problems
2014-2015 Spring OMO #13
Let
A
B
C
ABC
A
BC
be a scalene triangle whose side lengths are positive integers. It is called stable if its three side lengths are multiples of 5, 80, and 112, respectively. What is the smallest possible side length that can appear in any stable triangle?Proposed by Evan Chen
2015-2016 Fall OMO #13
You live in an economy where all coins are of value
1
/
k
1/k
1/
k
for some positive integer
k
k
k
(i.e.
1
,
1
/
2
,
1
/
3
,
…
1, 1/2, 1/3, \dots
1
,
1/2
,
1/3
,
…
). You just recently bought a coin exchanging machine, called the Cape Town Machine . For any integer
n
>
1
n > 1
n
>
1
, this machine can take in
n
n
n
of your coins of the same value, and return a coin of value equal to the sum of values of those coins (provided the coin returned is part of the economy). Given that the product of coins values that you have is
201
5
−
1000
2015^{-1000}
201
5
−
1000
, what is the maximum numbers of times you can use the machine over all possible starting sets of coins? Proposed by Yang Liu
12
2
Hide problems
2014-2015 Spring OMO #12
At the Intergalactic Math Olympiad held in the year 9001, there are 6 problems, and on each problem you can earn an integer score from 0 to 7. The contestant's score is the product of the scores on the 6 problems, and ties are broken by the sum of the 6 problems. If 2 contestants are still tied after this, their ranks are equal. In this olympiad, there are
8
6
=
262144
8^6=262144
8
6
=
262144
participants, and no two get the same score on every problem. Find the score of the participant whose rank was
7
6
=
117649
7^6 = 117649
7
6
=
117649
.Proposed by Yang Liu
2015-2016 Fall OMO #12
Let
a
a
a
,
b
b
b
,
c
c
c
be the distinct roots of the polynomial
P
(
x
)
=
x
3
−
10
x
2
+
x
−
2015
P(x) = x^3 - 10x^2 + x - 2015
P
(
x
)
=
x
3
−
10
x
2
+
x
−
2015
. The cubic polynomial
Q
(
x
)
Q(x)
Q
(
x
)
is monic and has distinct roots
b
c
−
a
2
bc-a^2
b
c
−
a
2
,
c
a
−
b
2
ca-b^2
c
a
−
b
2
,
a
b
−
c
2
ab-c^2
ab
−
c
2
. What is the sum of the coefficients of
Q
Q
Q
?Proposed by Evan Chen
11
2
Hide problems
2014-2015 Spring OMO #11
Let
S
S
S
be a set. We say
S
S
S
is
D
∗
D^\ast
D
∗
-finite if there exists a function
f
:
S
→
S
f : S \to S
f
:
S
→
S
such that for every nonempty proper subset
Y
⊊
S
Y \subsetneq S
Y
⊊
S
, there exists a
y
∈
Y
y \in Y
y
∈
Y
such that
f
(
y
)
∉
Y
f(y) \notin Y
f
(
y
)
∈
/
Y
. The function
f
f
f
is called a witness of
S
S
S
. How many witnesses does
{
0
,
1
,
⋯
,
5
}
\{0,1,\cdots,5\}
{
0
,
1
,
⋯
,
5
}
have?Proposed by Evan Chen
2015-2016 Fall OMO #11
A trapezoid
A
B
C
D
ABCD
A
BC
D
lies on the
x
y
xy
x
y
-plane. The slopes of lines
B
C
BC
BC
and
A
D
AD
A
D
are both
1
3
\frac 13
3
1
, and the slope of line
A
B
AB
A
B
is
−
2
3
-\frac 23
−
3
2
. Given that
A
B
=
C
D
AB=CD
A
B
=
C
D
and
B
C
<
A
D
BC< AD
BC
<
A
D
, the absolute value of the slope of line
C
D
CD
C
D
can be expressed as
m
n
\frac mn
n
m
, where
m
,
n
m,n
m
,
n
are two relatively prime positive integers. Find
100
m
+
n
100m+n
100
m
+
n
. Proposed by Yannick Yao
10
2
Hide problems
2014-2015 Spring OMO #10
Nicky has a circle. To make his circle look more interesting, he draws a regular 15-gon, 21-gon, and 35-gon such that all vertices of all three polygons lie on the circle. Let
n
n
n
be the number of distinct vertices on the circle. Find the sum of the possible values of
n
n
n
.Proposed by Yang Liu
2015-2016 Fall OMO #10
For any positive integer
n
n
n
, define a function
f
f
f
by
f
(
n
)
=
2
n
+
1
−
2
⌊
log
2
n
⌋
+
1
.
f(n)=2n+1-2^{\lfloor\log_2n\rfloor+1}.
f
(
n
)
=
2
n
+
1
−
2
⌊
l
o
g
2
n
⌋
+
1
.
Let
f
m
f^m
f
m
denote the function
f
f
f
applied
m
m
m
times.. Determine the number of integers
n
n
n
between
1
1
1
and
65535
65535
65535
inclusive such that
f
n
(
n
)
=
f
2015
(
2015
)
.
f^n(n)=f^{2015}(2015).
f
n
(
n
)
=
f
2015
(
2015
)
.
Proposed by Yannick Yao
9
2
Hide problems
2014-2015 Spring OMO #9
Find the sum of the decimal digits of the number
5
∑
k
=
1
99
k
(
k
+
1
)
(
k
2
+
k
+
1
)
.
5\sum_{k=1}^{99} k(k + 1)(k^2 + k + 1).
5
k
=
1
∑
99
k
(
k
+
1
)
(
k
2
+
k
+
1
)
.
Proposed by Robin Park
2015-2016 Fall OMO #9
Let
s
1
,
s
2
,
…
s_1, s_2, \dots
s
1
,
s
2
,
…
be an arithmetic progression of positive integers. Suppose that s_{s_1} = x+2, s_{s_2} = x^2+18, \text{and} s_{s_3} = 2x^2+18. Determine the value of
x
x
x
. Proposed by Evan Chen
8
2
Hide problems
2014-2015 Spring OMO #8
Determine the number of sequences of positive integers
1
=
x
0
<
x
1
<
⋯
<
x
10
=
1
0
5
1 = x_0 < x_1 < \dots < x_{10} = 10^{5}
1
=
x
0
<
x
1
<
⋯
<
x
10
=
1
0
5
with the property that for each
m
=
0
,
…
,
9
m=0,\dots,9
m
=
0
,
…
,
9
the number
x
m
+
1
x
m
\tfrac{x_{m+1}}{x_m}
x
m
x
m
+
1
is a prime number.Proposed by Evan Chen
2015-2016 Fall OMO #8
The two numbers
0
0
0
and
1
1
1
are initially written in a row on a chalkboard. Every minute thereafter, Denys writes the number
a
+
b
a+b
a
+
b
between all pairs of consecutive numbers
a
a
a
,
b
b
b
on the board. How many odd numbers will be on the board after
10
10
10
such operations?Proposed by Michael Kural
7
2
Hide problems
2014-2015 Spring OMO #7
A geometric progression of positive integers has
n
n
n
terms; the first term is
1
0
2015
10^{2015}
1
0
2015
and the last term is an odd positive integer. How many possible values of
n
n
n
are there?Proposed by Evan Chen
2015-2016 Fall OMO #7
Define sequence
{
a
n
}
\{a_n\}
{
a
n
}
as following:
a
0
=
0
a_0=0
a
0
=
0
,
a
1
=
1
a_1=1
a
1
=
1
, and
a
i
=
2
a
i
−
1
−
a
i
−
2
+
2
a_{i}=2a_{i-1}-a_{i-2}+2
a
i
=
2
a
i
−
1
−
a
i
−
2
+
2
for all
i
≥
2
i\geq 2
i
≥
2
. Determine the value of
a
1000
.
a_{1000}.
a
1000
.
Proposed by Yannick Yao
6
2
Hide problems
2014-2015 Spring OMO #6
We delete the four corners of a
8
×
8
8 \times 8
8
×
8
chessboard. How many ways are there to place eight non-attacking rooks on the remaining squares?Proposed by Evan Chen
2015-2016 Fall OMO #6
Farmer John has a (flexible) fence of length
L
L
L
and two straight walls that intersect at a corner perpendicular to each other. He knows that if he doesn't use any walls, he call enclose an maximum possible area of
A
0
A_0
A
0
, and when he uses one of the walls or both walls, he gets a maximum of area of
A
1
A_1
A
1
and
A
2
A_2
A
2
respectively. If
n
=
A
1
A
0
+
A
2
A
1
n=\frac{A_1}{A_0}+\frac{A_2}{A_1}
n
=
A
0
A
1
+
A
1
A
2
, find
⌊
1000
n
⌋
\lfloor 1000n\rfloor
⌊
1000
n
⌋
.Proposed by Yannick Yao
5
2
Hide problems
2014-2015 Spring OMO #5
Let
A
B
C
ABC
A
BC
be an isosceles triangle with
∠
A
=
9
0
∘
\angle A = 90^{\circ}
∠
A
=
9
0
∘
. Points
D
D
D
and
E
E
E
are selected on sides
A
B
AB
A
B
and
A
C
AC
A
C
, and points
X
X
X
and
Y
Y
Y
are the feet of the altitudes from
D
D
D
and
E
E
E
to side
B
C
BC
BC
. Given that
A
D
=
48
2
AD = 48\sqrt2
A
D
=
48
2
and
A
E
=
52
2
AE = 52\sqrt2
A
E
=
52
2
, compute
X
Y
XY
X
Y
.Proposed by Evan Chen
2015-2016 Fall OMO #5
Merlin wants to buy a magical box, which happens to be an
n
n
n
-dimensional hypercube with side length
1
1
1
cm. The box needs to be large enough to fit his wand, which is
25.6
25.6
25.6
cm long. What is the minimal possible value of
n
n
n
? Proposed by Evan Chen
4
2
Hide problems
2014-2015 Spring OMO #4
Find the sum of all distinct possible values of
x
2
−
4
x
+
100
x^2-4x+100
x
2
−
4
x
+
100
, where
x
x
x
is an integer between 1 and 100, inclusive.Proposed by Robin Park
2015-2016 Fall OMO #4
Let
ω
\omega
ω
be a circle with diameter
A
B
AB
A
B
and center
O
O
O
. We draw a circle
ω
A
\omega_A
ω
A
through
O
O
O
and
A
A
A
, and another circle
ω
B
\omega_B
ω
B
through
O
O
O
and
B
B
B
; the circles
ω
A
\omega_A
ω
A
and
ω
B
\omega_B
ω
B
intersect at a point
C
C
C
distinct from
O
O
O
. Assume that all three circles
ω
\omega
ω
,
ω
A
\omega_A
ω
A
,
ω
B
\omega_B
ω
B
are congruent. If
C
O
=
3
CO = \sqrt 3
CO
=
3
, what is the perimeter of
△
A
B
C
\triangle ABC
△
A
BC
?Proposed by Evan Chen
3
2
Hide problems
2014-2015 Spring OMO #3
On a large wooden block there are four twelve-hour analog clocks of varying accuracy. At 7PM on April 3, 2015, they all correctly displayed the time. The first clock is accurate, the second clock is two times as fast as the first clock, the third clock is three times as fast as the first clock, and the last clock doesn't move at all. How many hours must elapse (from 7PM) before the times displayed on the clocks coincide again? (The clocks do not distinguish between AM and PM.) [asy] import olympiad; import cse5;size(12cm); defaultpen(linewidth(0.9)+fontsize(11pt));picture clock(real hh, real mm, string nn) { picture p; draw(p, unitcircle); for(int i=1;i<=12;i=i+1) { // draw(p, 0.9*dir(90-30*i)--dir(90-30*i)); label(p, "
"
+
(
s
t
r
i
n
g
)
i
+
"
"+(string) i+"
"
+
(
s
t
r
in
g
)
i
+
"
",0.84*dir(90-30*i), fontsize(9pt)); } dot(p, origin); pair hpoint = 0.5 * dir(90 - 30 * (hh + mm/60)); pair mpoint = 0.75 * dir(90 - 6 * mm); draw(p, origin--hpoint, EndArrow(HookHead, 3)); draw(p, origin--mpoint, EndArrow(HookHead, 5)); string tlabel; if (mm > 10) { tlabel = (string) hh + ":" + (string) mm; } else { tlabel = (string) hh + ":0" + (string) mm; } label(p, tlabel, dir(90)*1.2, dir(90)); label(p, tlabel, dir(90)*1.2, dir(90)); label(p, nn, dir(-90)*1.1, dir(-90)); return p; }// The block real h = 1; filldraw( (-1.2,-1)--(8.4,-1)--(8.4,-1-h)--(-1.2,-1-h)--cycle, 0.7*lightgrey, black);add(shift((0.0,0)) * clock(10,22, "I")); add(shift((2.4,0)) * clock( 1,44, "II")); add(shift((4.8,0)) * clock( 5,06, "III")); add(shift((7.2,0)) * clock( 7,00, "IV"));label("\emph{Omnes vulnerant, postuma necat}", (3.6, -1.8), origin); [/asy]Proposed by Evan Chen
2015-2016 Fall OMO #3
How many integers between 123 and 321 inclusive have exactly two digits that are 2? Proposed by Yannick Yao
2
2
Hide problems
2014-2015 Spring OMO #2
A classroom has
30
30
30
students, each of whom is either male or female. For every student
S
S
S
, we define his or her ratio to be the number of students of the opposite gender as
S
S
S
divided by the number of students of the same gender as
S
S
S
(including
S
S
S
). Let
Σ
\Sigma
Σ
denote the sum of the ratios of all
30
30
30
students. Find the number of possible values of
Σ
\Sigma
Σ
.Proposed by Evan Chen
2015-2016 Fall OMO #2
At a national math contest, students are being housed in single rooms and double rooms; it is known that
75
%
75\%
75%
of the students are housed in double rooms. What percentage of the rooms occupied are double rooms?Proposed by Evan Chen
1
2
Hide problems
2014-2015 Spring OMO #1
What is the largest positive integer which is equal to the sum of its digits?Proposed by Evan Chen
2015-2016 Fall OMO #1
Evaluate
(
8
2
)
+
(
9
2
)
+
(
15
2
)
+
(
16
2
)
.
\sqrt{\binom82+\binom92+\binom{15}2+\binom{16}2}.
(
2
8
)
+
(
2
9
)
+
(
2
15
)
+
(
2
16
)
.
Proposed by Evan Chen